\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{8}   &  -34.0 & +  8.00 x_{1} & +  9.00 x_{2} & -10.00 x_{3} & -7.00 x_{4} & +  8.00 x_{5} & -7.00 x_{6} & +  7.00 x_{7}\\
 x_{9}   &  30.0 & -4.00 x_{1} & -3.00 x_{2} & +  8.00 x_{3} & -10.00 x_{4} & -1.00 x_{5} & +  8.00 x_{6} & -8.00 x_{7}\\
 x_{10}   &  -16.0 & +  1.00 x_{1} & -1.00 x_{2} & -4.00 x_{3} & -9.00 x_{4} & +  9.00 x_{5} & +  7.00 x_{6} & -1.00 x_{7}\\
 x_{11}   &  -2.0 & -8.00 x_{1} & -1.00 x_{2} & -7.00 x_{3} & -1.00 x_{4} & -4.00 x_{5} & +  5.00 x_{6} & +  4.00 x_{7}\\
 x_{12}   &  -16.0 & +  3.00 x_{1} & +  7.00 x_{2} & -3.00 x_{3} & +  4.00 x_{4} &   & +  9.00 x_{6} & +  1.00 x_{7}\\
 x_{13}   &  6.0 & -9.00 x_{1} & -8.00 x_{2} &   & +  2.00 x_{4} & +  1.00 x_{5} & +  9.00 x_{6} &   \\
\hline
z    &  0.0  &   & -4.00 x_{2} & +  2.00 x_{3} & -2.00 x_{4} & -3.00 x_{5} & -3.00 x_{6} & -5.00 x_{7}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{7}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,13$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -8.00 y_{8} & +  4.00 y_{9} & -1.00 y_{10} & +  8.00 y_{11} & -3.00 y_{12} & +  9.00 y_{13}\\
 y_{2}   &  1.0 & -9.00 y_{8} & +  3.00 y_{9} & +  1.00 y_{10} & +  1.00 y_{11} & -7.00 y_{12} & +  8.00 y_{13}\\
 y_{3}   &  1.0 & + 10.00 y_{8} & -8.00 y_{9} & +  4.00 y_{10} & +  7.00 y_{11} & +  3.00 y_{12} &   \\
 y_{4}   &  1.0 & +  7.00 y_{8} & + 10.00 y_{9} & +  9.00 y_{10} & +  1.00 y_{11} & -4.00 y_{12} & -2.00 y_{13}\\
 y_{5}   &  1.0 & -8.00 y_{8} & +  1.00 y_{9} & -9.00 y_{10} & +  4.00 y_{11} &   & -1.00 y_{13}\\
 y_{6}   &  1.0 & +  7.00 y_{8} & -8.00 y_{9} & -7.00 y_{10} & -5.00 y_{11} & -9.00 y_{12} & -9.00 y_{13}\\
 y_{7}   &  1.0 & -7.00 y_{8} & +  8.00 y_{9} & +  1.00 y_{10} & -4.00 y_{11} & -1.00 y_{12} &   \\
\hline
z    &  -0 & + 34.00 y_{8} & -30.00 y_{9} & + 16.00 y_{10} & +  2.00 y_{11} & + 16.00 y_{12} & -6.00 y_{13}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 6 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  0.886626199334 & -0.34 y_{6} & +  5.32 y_{9} & +  0.39 y_{5} & -0.93 y_{7} & +  1.00 y_{2} & -1.74 y_{13}\\
 y_{8}   &  0.086547875465 & +  0.02 y_{6} & +  0.77 y_{9} & -0.03 y_{5} & -0.06 y_{7} & -0.02 y_{2} & +  0.35 y_{13}\\
 y_{3}   &  3.09046406893 & -0.32 y_{6} & +  2.20 y_{9} & -0.35 y_{5} & -1.64 y_{7} & +  0.22 y_{2} & -5.05 y_{13}\\
 y_{4}   &  2.20246720188 & -0.06 y_{6} & + 15.78 y_{9} & -0.97 y_{5} & -0.95 y_{7} & +  0.78 y_{2} & -9.77 y_{13}\\
 y_{10}   &  0.0804777756021 & -0.04 y_{6} & -0.25 y_{9} & -0.08 y_{5} & -0.02 y_{7} & +  0.05 y_{2} & -0.84 y_{13}\\
 y_{12}   &  0.0579596632074 & -0.04 y_{6} & -0.49 y_{9} & +  0.03 y_{5} & +  0.05 y_{7} & -0.10 y_{2} & +  0.44 y_{13}\\
 y_{11}   &  0.104170746035 & -0.04 y_{6} & +  0.72 y_{9} & +  0.02 y_{5} & -0.16 y_{7} & +  0.07 y_{2} & -0.93 y_{13}\\
\hline
z    &  5.36596827883 & -0.56 y_{6} & -14.39 y_{9} & -1.69 y_{5} & -1.81 y_{7} & -1.31 y_{2} & -2.30 y_{13}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  0.561386332485 & +  0.34 x_{1} & -0.02 x_{8} & +  0.32 x_{3} & +  0.06 x_{4} & +  0.04 x_{10} & +  0.04 x_{12} & +  0.04 x_{11}\\
 x_{9}   &  14.392206775 & -5.32 x_{1} & -0.77 x_{8} & -2.20 x_{3} & -15.78 x_{4} & +  0.25 x_{10} & +  0.49 x_{12} & -0.72 x_{11}\\
 x_{5}   &  1.6874877619 & -0.39 x_{1} & +  0.03 x_{8} & +  0.35 x_{3} & +  0.97 x_{4} & +  0.08 x_{10} & -0.03 x_{12} & -0.02 x_{11}\\
 x_{7}   &  1.81202271392 & +  0.93 x_{1} & +  0.06 x_{8} & +  1.64 x_{3} & +  0.95 x_{4} & +  0.02 x_{10} & -0.05 x_{12} & +  0.16 x_{11}\\
 x_{2}   &  1.30507147053 & -1.00 x_{1} & +  0.02 x_{8} & -0.22 x_{3} & -0.78 x_{4} & -0.05 x_{10} & +  0.10 x_{12} & -0.07 x_{11}\\
 x_{13}   &  2.29939299001 & +  1.74 x_{1} & -0.35 x_{8} & +  5.05 x_{3} & +  9.77 x_{4} & +  0.84 x_{10} & -0.44 x_{12} & +  0.93 x_{11}\\
\hline
z    &  -5.36596827883 & -0.89 x_{1} & -0.09 x_{8} & -3.09 x_{3} & -2.20 x_{4} & -0.08 x_{10} & -0.06 x_{12} & -0.10 x_{11}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  0.561386332485 & +  0.34 x_{1} & -0.02 x_{8} & +  0.32 x_{3} & +  0.06 x_{4} & +  0.04 x_{10} & +  0.04 x_{12} & +  0.04 x_{11}\\
 x_{9}   &  14.392206775 & -5.32 x_{1} & -0.77 x_{8} & -2.20 x_{3} & -15.78 x_{4} & +  0.25 x_{10} & +  0.49 x_{12} & -0.72 x_{11}\\
 x_{5}   &  1.6874877619 & -0.39 x_{1} & +  0.03 x_{8} & +  0.35 x_{3} & +  0.97 x_{4} & +  0.08 x_{10} & -0.03 x_{12} & -0.02 x_{11}\\
 x_{7}   &  1.81202271392 & +  0.93 x_{1} & +  0.06 x_{8} & +  1.64 x_{3} & +  0.95 x_{4} & +  0.02 x_{10} & -0.05 x_{12} & +  0.16 x_{11}\\
 x_{2}   &  1.30507147053 & -1.00 x_{1} & +  0.02 x_{8} & -0.22 x_{3} & -0.78 x_{4} & -0.05 x_{10} & +  0.10 x_{12} & -0.07 x_{11}\\
 x_{13}   &  2.29939299001 & +  1.74 x_{1} & -0.35 x_{8} & +  5.05 x_{3} & +  9.77 x_{4} & +  0.84 x_{10} & -0.44 x_{12} & +  0.93 x_{11}\\
\hline
z    &  -21.0270217349 & -0.52 x_{1} & -0.40 x_{8} & -7.33 x_{3} & -6.73 x_{4} & -0.22 x_{10} & -0.16 x_{12} & -0.56 x_{11}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  0.561386332485 & +  0.34 x_{1} & -0.02 x_{8} & +  0.32 x_{3} & +  0.06 x_{4} & +  0.04 x_{10} & +  0.04 x_{12} & +  0.04 x_{11}\\
 x_{9}   &  14.392206775 & -5.32 x_{1} & -0.77 x_{8} & -2.20 x_{3} & -15.78 x_{4} & +  0.25 x_{10} & +  0.49 x_{12} & -0.72 x_{11}\\
 x_{5}   &  1.6874877619 & -0.39 x_{1} & +  0.03 x_{8} & +  0.35 x_{3} & +  0.97 x_{4} & +  0.08 x_{10} & -0.03 x_{12} & -0.02 x_{11}\\
 x_{7}   &  1.81202271392 & +  0.93 x_{1} & +  0.06 x_{8} & +  1.64 x_{3} & +  0.95 x_{4} & +  0.02 x_{10} & -0.05 x_{12} & +  0.16 x_{11}\\
 x_{2}   &  1.30507147053 & -1.00 x_{1} & +  0.02 x_{8} & -0.22 x_{3} & -0.78 x_{4} & -0.05 x_{10} & +  0.10 x_{12} & -0.07 x_{11}\\
 x_{13}   &  2.29939299001 & +  1.74 x_{1} & -0.35 x_{8} & +  5.05 x_{3} & +  9.77 x_{4} & +  0.84 x_{10} & -0.44 x_{12} & +  0.93 x_{11}\\
\hline
z    &  -21.0270217349 & -0.52 x_{1} & -0.40 x_{8} & -7.33 x_{3} & -6.73 x_{4} & -0.22 x_{10} & -0.16 x_{12} & -0.56 x_{11}\\
\end{array}\]
Final Dictionary
Solution:  -21.0270217349
Num Pivots:  1
\end{document}
